In the introduction the paper gives the known results for the ordinary incomplete gamma functions, but the main interest is a class of generalized incomplete gamma functions, and to derive uniform asymptotic expansions. Let be a positive integer. Then the Mellin integral
can be viewed as a normalized exponential function, with . Inversion gives Then the function is introduced as the incomplete gamma function of order . The problem is to find an asymptotic expansion as , which holds uniformly with respect to x. Further generalizations of the exponential and incomplete gamma functions are also considered. The research is motivated by the theory of general Dirichlet series, where the generalized incomplete gamma functions arise.